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Dependence and Correlation

In everyday life I hear the word "correlation" thrown around far more often than "dependence." What's the difference? Correlation, in its most common form, is a measure of linear dependence; the catch is that not all dependencies are linear. The set of correlated random variables lies entirely within of the larger set of dependent random variables; correlation implies dependence, but not the other way around. Here are some silly (but hopefully interesting) examples to illustrate that point:

The last two are classic examples: X and Y are normally distributed, but (X, Y) is not a bivariate normal.

I'll admit that the two exponentials are a bit counterintuitive to me, at least visually. (They're in the second plot from the top, which looks vaguely like a B-2.) The variables are independent; if you regressed Y on X you'd end up with a flat line. Yet, somehow, if I were to look at that plot without knowing how the variables were generated, I'd want to draw a diagonal line pointing up and to the right. If anything, it goes to show that I should probably not run regressions "by inspection."

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